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Abstract
Let X be a compact
2 -manifold with nonempty
boundary ∂ X and let
f : ( X , ∂ X ) → ( X , ∂ X ) be a boundary-preserving
map. Denote by M F ∂ [ f ]
the minimum number of fixed point among all boundary-preserving
maps that are homotopic through boundary-preserving maps to
f . The relative
Nielsen number N ∂ ( f )
is the sum of the number of essential fixed point classes of the restriction
f ̄ : ∂ X → ∂ X
and the number of essential fixed point classes of
f
that do not contain essential fixed point classes of
f ̄ . We prove
that if X
is the Möbius band with one (open) disc removed, then
M F ∂ [ f ] − N ∂ ( f ) ≤ 1 for all maps
f : ( X , ∂ X ) → ( X , ∂ X ) . This result
is the final step in the boundary-Wecken classification of surfaces, which is as follows. If
X is the disc, annulus or
Möbius band, then X is
boundary-Wecken, that is, M F ∂ [ f ] = N ∂ ( f ) for
all boundary-preserving maps. If X
is the disc with two discs removed or the Möbius band with one disc removed, then
X is not
boundary-Wecken, but M F ∂ [ f ] − N ∂ ( f ) ≤ 1 .
All other surfaces are totally non-boundary-Wecken, that is, given an integer
k ≥ 1 , there
is a map f k : ( X , ∂ X ) → ( X , ∂ X )
such that M F ∂ [ f k ] − N ∂ ( f k ) ≥ k .
Keywords
boundary-Wecken, relative Nielsen number, punctured Möbius
band, boundary-preserving map
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 54H25, 57N05
Publication
Received: 21 November 2002
Revised: 15 October 2003
Accepted: 26 November 2003
Published: 7 February 2004